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Transactions of the American Mathematical Society

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On the quadratic dual of the Fomin-Kirillov algebras


Authors: Chelsea Walton and James J. Zhang
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 16W50, 16P40, 16P90, 16E65
DOI: https://doi.org/10.1090/tran/7781
Published electronically: February 11, 2019
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Abstract: We study ring-theoretic and homological properties of the quadratic dual (or Koszul dual) $ \mathcal {E}_n^!$ of the Fomin-Kirillov algebras $ \mathcal {E}_n$; these algebras are connected $ \mathbb{N}$-graded and are defined for $ n \geq 2$. We establish that the algebra $ \mathcal {E}_n^!$ is module finite over its center (and thus satisfies a polynomial identity), is Noetherian, and has Gelfand-Kirillov dimension $ \lfloor n/2 \rfloor $ for each $ n \geq 2$. We also observe that $ \mathcal {E}_n^!$ is not prime for $ n \geq 3$. By a result of Roos, $ \mathcal {E}_n$ is not Koszul for $ n \geq 3$, so neither is $ \mathcal {E}_n^!$ for $ n \geq 3$. Nevertheless, we prove that $ \mathcal {E}_n^!$ is Artin-Schelter (AS-)regular if and only if $ n=2$, and that $ \mathcal {E}_n^!$ is both AS-Gorenstein and AS-Cohen-Macaulay if and only if $ n=2,3$. We also show that the depth of $ \mathcal {E}_n^!$ is $ \leq 1$ for each $ n \geq 2$, conjecture that we have equality, and show that this claim holds for $ n =2,3$. Several other directions for further examination of $ \mathcal {E}_n^!$ are suggested at the end of this article.


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Additional Information

Chelsea Walton
Affiliation: Department of Mathematics, The University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
Email: notlaw@illinois.edu

James J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: zhang@math.washington.edu

DOI: https://doi.org/10.1090/tran/7781
Keywords: Fomin--Kirillov algebra, Gelfand--Kirillov dimension, homological conditions, quadratic (Koszul) dual, Noetherian, depth
Received by editor(s): July 6, 2018
Published electronically: February 11, 2019
Additional Notes: The first author was partially supported by a research fellowship from the Alfred P. Sloan foundation, and by the U.S. National Science Foundation grants #DMS-1663775, 1903192. This work was completed during her visits to the University of Washington–Seattle.
The second author was partially supported by U.S. National Science Foundation grant #DMS-1700825.
Part of this work was completed during the authors’ attendance at the “Quantum Homogeneous Spaces” workshop at the International Centre for Mathematical Sciences in Edinburgh, Scotland; the authors appreciate the institution staff for their hospitality and assistance during these stays.
Article copyright: © Copyright 2019 American Mathematical Society